Riemannian geometry of half lightlike

manifolds

K. L. DUGGAL

Abstract. We study a class of lightlike manifolds M , called half light-

like submanifold of codimension 2 of a Minkowski spacetime Rn+21 . We

show that some specified aspects of the null geometry of M reduce to

the corresponding Riemannian geometry of its spacelike hypersurface.

1. Introduction

Since the middle of the twentieth century Riemannian geometry hascreated a substantial influence on several main areas of mathemati-cal sciences. For example, see Berger’s recent survey book [2], withvoluminous bibliography. Primarily, semi-Riemannian (in particu-lar, Lorentzian [1]) geometry has its roots in Riemannian geometry.On the other hand, the situation is quite different for lightlike man-ifolds with a degenerate metric, since one fails to use, in the usualway, the theory of non-null geometry. Due to growing importance ofnull geometry in mathematical physics and limited information onits geometric theory it is reasonable to ask the following question:

2002 Mathematics Subject Classification. 53C20, 53C50.

Key words. Lightlike submanifold, Riemannian geometry.

1

Find a technique which reduces the problems of lightlike geome-try to problems of Riemannian or semi - Riemannian geometry.

The subject of this paper is to answer above question, in the affir-mative, for a particular class of lightlike submanifolds (M, g), calledhalf - lightlike submanifolds ( see [5, 6]), of a Minkowski spacetime(Rn+2

1 , g). In Section 2, we recall those results on half - lightlikesubmanifolds which we have used in this paper. In Section 3, weshow that M is a product manifold L × M ′, where L and M ′ are a1-dimensional null space and an (n − 1) - dimensional Riemanniansubmanifold of Rn+2

1 . Based on this, we prove the main result (seeTheorem 4) of this paper.

2. Half Lightlike Manifolds

Let ( M, g ) be an (n+ 2) - dimensional semi - Riemannian manifoldof constant index q ≥ 1 and M a submanifold of codimension 2of M . In case g is degenerate on the tangent bundle TM of Mwe say that M is a lightlike (null) submanifold of M . Denote byF (M) the algebra of smooth functions on M and by Γ(E) the F (M)module of smooth sections of a vector bundle E over M . We usethe same notation for any other vector bundle. All manifolds areparacompact and smooth. Denote by g the induced tensor field ofg on M and suppose g is degenerate, that is, there exists locally avector field ξ ∈ Γ (TM), ξ 6= 0, such that g ( ξ , X ) = 0, for anyX ∈ Γ (TM). Then, for each tangent space TxM we consider

TxM⊥ = U ∈ Tx M : g (U , V ) = 0, ∀V ∈ TxM

which is a degenerate 2 - dimensional subspace of Tx M . Since M islightlike, both TxM and TxM

⊥ are degenerate orthogonal subspacesbut no longer complementary. Denote by RadTM a radical (null)distribution of the tangent bundle space TM of M . In this casedim(RadTxM) = TxM

⋂TxM

⊥ depends on the point x ∈ M. Inthis paper we study half-lightlike submanifolds defined as follows:

Definition 1. [5] An n - dimensional submanifold M of M is calleda half - lightlike submanifold of codimension 2 if dim(RadTM) = 1,n ≥ 2, and there exist ξ1 , ξ2 ∈ TxM⊥ such that

g ( ξ1, U ) = 0, g ( ξ2 , ξ2 ) 6= 0, ∀U ∈ TxM⊥.

Above relations imply that ξ1 ∈ RadTxM . Thus, locally thereexists a lightlike vector field on M , also denoted by ξ1, such that

g ( ξ1 , X ) = g ( ξ1 , W ) = 0, ∀X ∈ Γ(TM), W ∈ Γ(TM⊥)

and RadTM is locally spanned by ξ1. In this case there exists anon-degenerate screen distribution S(TM) which is supplementaryto RadTM in TM [5]. Thus,

TM = RadTM ⊕ S(TM). (1)

Although S(TM) is not unique, it is canonically isomorphic tothe factor vector bundle TM /RadTM . The existence of S(TM)is secured for a paracompact lightlike manifold M . Our results willbe based on a specified choice of S(TM).

Consider the orthogonal complementary distribution S(TM)⊥

to SM in TM . Certainly ξ1 and ξ2 belong to Γ (S(TM)⊥ ). Wechoose ξ2 as a unit vector field and put g ( ξ2 , ξ2 ) = ε = ± 1. SinceRadTM is a 1 - dimensional vector sub bundle of TM⊥, we considera supplementary distribution D to RadTM such that it is locallyrepresented by ξ2. Hence we have the decomposition

S(TM)⊥ = D ⊥ D⊥,

where D⊥ is the orthogonal complementary distribution to D inSM⊥. Taking into account that the fibers ofD⊥ are non - degenerateplanes and ξ1 ∈ Γ (D⊥ ), there exists a unique locally defined vectorfield N ∈ Γ (D⊥ ), satisfying

g (N , ξ1 ) 6= 0, g (N , N ) = g (N , ξ2 ) = 0.

Hence N is another lightlike vector field which is neither tangent toM nor collinear with ξ2. Define a vector bundle tr(TM) over M by

tr(TM) = D ⊕ ntr(TM),

where ntr(TM) is a 1 - dimensional vector bundle locally repre-sented by N , and is called a canonical affine normal bundle of Mwith respect to SM . Therefore,

TM = RadTM ⊕ S(TM) ⊕ tr(TM)

= RadTM ⊕ S(TM) ⊕ D ⊕ ntr(TM). (2)

Choose the frames ξ1,W1, ...,Wn−1 and ξ1,W1, ..., Wn−1, ξ2, Non M and M , where W1 , ... , Wn− 1 is a basis of Γ(S(TM)).

Example 1. Let (M, g) be a 2 - surface of (R41 , g ), defined by

xA = xA (u , v) , A ∈ 0, 1, 2, 3, with the metric g given byg (x , y) = −x0 y0 + x1 y1 + x2 y2 + x3 y3. Then, TM is spanned by

∂

∂ u=

∂ xA

∂ u

∂

∂ xA;

∂

∂ v=

∂ xA

∂ v

∂

∂ xA

.

Denote by

DAB =

∣∣∣∣∣ ∂ xA

∂ u∂ xB

∂ u∂ xA

∂ v∂ xB

∂ v

∣∣∣∣∣ .It is easy to show (see [6, pages 155-156]) that M is lightlike if

and only if on each coordinate neighborhood U ⊂ M we have

3∑a=1

(Doa)2 =∑

1≤a<b≤3

(Dab)2.

Then, by direct calculations we find that M is a half-lightlike surfacewhere RadTM , ntr (TM) , S(TM) and D are spanned by

ξ1 = ξA∂

∂ xA, ξA =

3∑B=0

εBDAB ∂ x

B

∂ u,

N =1

2(ξ0)2

(− ξ0 ∂

∂ x0+

3∑a=1

ξa∂

∂ xa

),

V =∂ x0

∂ v

∂

∂ u− ∂ x0

∂ u

∂

∂ v=

3∑a=1

Dao ∂

∂xa,

U = D23 ∂

∂ x1+D31 ∂

∂ x2+D12 ∂

∂ x3

respectively. Here εB is the signature of the basis

∂∂ xB

with

respect to the Minkowski metric g. Summing up, we obtain a quasi -orthonormal frames field

ξ1 , W =1√4V , N , ξ2 =

1√4U , 4 ≡

3∑a=1

(Da0)2.

In particular, let M be given by the equations

x0 = u cosh v + sinh v , x1 = u+ v , x2 = u sinh v + cosh v , x3 = v.

Then, by direct calculations we get 4 = (1 + u2) cosh2 v and

D10 = u sinh v , D20 = −u , D30 = − cosh v

D23 = sinh v D31 = − 1 , D12 = u cosh v,

ξ1 = ( cosh v , 1 , sinh v , 0 ),

N =1

2 cosh2 v(− cosh v , 1 , sinh v , 0 ),

W =1√4

( 0 , u sinh v , −u , − cosh v ),

ξ2 =1√4

( 0 , sinh v , − 1 , u cosh v ).

Denote by P the projection of TM on S(TM) with respect tothe decomposition (2). Then,

X = PX + η(X) ξ1 , ∀X ∈ Γ(TM) ,

where η is a local differential 1 - form on M given by

η (X) = g(X , N). (3)

Let ∇ be the metric connection on M . Using (1) and (2), we put

∇XY = ∇XY + h (X , Y ) ,

∇XN = ∇XN − AN X , (4)

∇Xξ2 = ∇Xξ2 − Aξ2X ,

for any X , Y ∈ Γ (TM), where ∇XY , AN X and Aξ2X belong toΓ (TM), while h (X , Y ),∇X N and ∇Xξ2 belong to Γ (tr (TM)). Itis easy to check that∇ is a torsion - free linear connection on M , h isa Γ (tr (TM)) - valued symmetric F (M) - bilinear form on Γ (TM).Since ξ1 , N is locally a pair of lightlike sections on U ⊂ M ,we define symmetric F (M) - bilinear forms D1 and D2 and 1 - formsρ , τ , θ and ψ on U by

D1 (X, Y ) = g (h (X, Y ), ξ1 ) , D2 (X, Y ) = ε g (h (X, Y ), ξ2 ),

ρ (X) = g (∇XN , ξ1 ), τ (X) = ε g (∇XN , ξ2 ),

θ (X) = g (∇Xξ2 , ξ1 ), ψ (X) = ε g (∇Xξ2 , ξ2 ),

for any X , Y ∈ Γ (TM). It follows that

h (X , Y ) = D1 (X , Y )N +D2 (X , Y ) ξ2,

∇XN = ρ (X)N + τ (X) ξ2 , (5)

∇Xξ2 = θ (X)N + ψ (X) ξ2 .

Hence, on U , (4) become

∇XY = ∇XY +D1 (X , Y )N +D2 (X , Y )ξ2 , (6)

∇XN = −AN X + ρ (X)N + τ (X) ξ2 , (7)

∇Xξ2 = −Aξ2X + θ (X)N + ψ (X) ξ2 , (8)

for any X , Y ∈ Γ (TM). We call h, D1 and D2 the second funda-mental form, the lightlike second fundamental form and the screensecond fundamental form of M with respect to tr (TM) respectively.Following results easily follow from (6)- (8).

D1 (X , ξ1) = 0, ψ(X) = 0, g (AN X , N) = 0, (9)

g (Aξ2X , Y ) = εD2 (X , Y ) + θ (X) η (Y ). (10)

By using (3), (6) - (8) and (10), we obtain

ρ (X) = − η (∇Xξ1), (11)

τ (X) = ε η (Aξ2X), (12)

θ (X) = − εD2(X , ξ1), ∀X ∈ Γ (TM). (13)

Next, consider the decomposition (1) and obtain

∇XPY = ∇∗XPY + h∗ (X ,PY ),

∇Xξ1 = −Aξ1X +∇⊥Xξ1 (14)

for anyX , Y ∈ Γ (TM), where∇∗XPY andAξ1 belong to Γ (S(TM)),while h∗ (X ,PY ) and ∇⊥Xξ1 belong to Γ (RadTM). It follows that∇∗ and∇⊥ are linear connections on the screen and radical distribu-tion respectively, Aξ1 is a linear operator on Γ (TM), h∗ is bilinearform on Γ (TM)× Γ (S(TM)). Locally, we define on U

E1 (X , PY ) = g (h∗ (X , PY ) , N ),

ρ1 (X) = g (∇⊥Xξ1 , N ), ∀X , Y ∈ Γ (TM).

It follows that h∗ (X , PY ) = E (X , PY ) ξ1 and ∇⊥Xξ1 = ρ (X) ξ1,∀X, Y ∈ Γ (TM). Hence, on U , locally, (14) become

∇XPY = ∇∗XPY + E(X ,PY ) ξ1, (15)

∇Xξ1 = −Aξ1X + ρ1 (X) ξ1. (16)

We call h∗ and E the second fundamental form and the local secondfundamental form of SM with respect to Rad (TM). Also,

E (X , PY ) = g(AN X , PY ), (17)

D1 (X , PY ) = g(Aξ1X , PY ), (18)

ρ1 (X) = − ρ (X), (19)

for any X , Y ∈ Γ (TM). Hence (16) becomes

∇Xξ1 = −Aξ1X − ρ (X) ξ1. (20)

Theorem 1. [5] Let M be a half - lightlike submanifold of M ofcodimension 2. Then the following assertions are equivalent:

(1) The screen distribution SM is integrable.

(2) The second fundamental form of SM is symmetric on Γ (SM).

(3) The shape operator AN1 of the immersion of M in M is sym-metric with respect to g on Γ (SM).

Since∇ is a metric connection, using (6), we obtain

(∇Xg) (Y , Z) = D1(X , Y ) η (Z) +D1 (X ,Z) η (Y ), (21)

∀X , Y , Z ∈ Γ (TM). In particular, the following holds:

Theorem 2. [5] Let M be a half - lightlike submanifold of M ofcodimension 2. Then the following assertions are equivalent:

(1) The induced connection ∇ on M is a metric connection.

(2) D1 vanishes identically on M .

(3) Aξ1 vanishes identically on M .

(4) ξ1 is a Killing vector field.

(5) TM⊥ is a parallel distribution with respect to ∇.

3. Geometry of Half Lightlike Manifolds

Let (Rn+21 , g) be a Minkowski spacetime with Lorentz metric g

of signature (− + . . . + ) and local coordinates (x0 , . . . xn, xn+1 ).Consider an n - dimensional submanifold (M, g) of Rn+2

1 , defined by

xA = fA(u0, . . . , un−1) ; rank

[∂fA

∂uα

]= n ,

where A ∈ 0, . . . , n+ 1 , α ∈ 0, . . . , n− 1 and fA are smoothfunctions on a coordinate neighborhood U ⊂M . Set

DA =

∣∣∣∣∣∣∣∣∣∣∣∣

∂f0

∂u0... ∂f

A−1

∂u0∂fA+1

∂u0... ∂f

n+1

∂u0

. . . .

. . . .

. . . .∂f0

∂un−1 ... ∂fA−1

∂un−1∂fA+1

∂un−1 ... ∂fn+1

∂un−1

∣∣∣∣∣∣∣∣∣∣∣∣.

Proposition 1. An n-dimensional submanifold M of Rn+21 is light-

like, if and only if, on each U , functions fA satisfy

(D0)2 =n+1∑a= 1

(Da)2 . (22)

In this case, the distribution RadTM ⊂ TM⊥ is spanned by

ξ1 = D0 ∂

∂x0+

n+1∑a=1

(−1)a−1Da ∂

∂xa. (23)

Proof. The natural frames field on U is given by

∂

∂uα=∂fA

∂uα∂

∂xA, α ∈ 0, . . . , n− 1 .

Then ξ1, given by (23), belongs to Γ(TM⊥). Hence, M is lightlikeif and only if g(ξ1, ξ1) = 0, which is equivalent with (22).

Note that a vector field V = −D0 ∂∂x0

of TRn+21 is nowhere

tangent to M , since g(V, ξ1) = (D0)2. Choose a null vector field N ,given by

N = (D0)−2 V +1

2ξ1 . (24)

It follows that g (N, ξ1 ) = 1. Denote by ntr(TM) the vector bun-dle spanned by N . A procedure similar to example 1 can be followedto get a unit spacelike vector field ξ2 ∈ TM⊥ and an orthonormalbasis for Γ(SM). Thus, M is a half-lightlike submanifold of Rn+1

1

and equation (2) will hold with TM = T (Rn+11 ).

Theorem 3. A half-lightlike submanifold M of a Minkowski spaceRn+2

1 has an integrable screen distribution. Moreover, M = L ×M ′

is a product manifold, where L and M ′ are a 1-dimensional nullspace of RadTM and an (n − 1) - dimensional Riemannian sub-manifold of Rn+2

1 respectively.

Proof. Let X, Y ∈ Γ(S(TM)), where S(TM) is a screen distribu-tion on M . Since ∇ is the Levi-Civita connection on Rn+2

1 , usingGauss equation (6) and (24) we obtain

g([X, Y ], aN + b ξ2) = a(D0)−1 g

(∇XY − ∇YX,

∂

∂x0

)+b g

(∇XY − ∇YX, ξ2

)= − a(D0)−1

g

(X, ∇Y

∂

∂x0

)− g

(Y, ∇X

∂

∂x0

)

= +bg(X, ∇Y ξ2

)− g

(Y, ∇Xξ2

)= 0 + 0, a, b ∈ R,

where for the second expression we have used (10) and η(X) = 0for X ∈ Γ(S(TM)). Hence, [X, Y ] ∈ Γ(S(TM)), that is, S(TM) isintegrable. Using well-known theorem of Frobenius, let M ′ be theintegral manifold of S(TM). Now RadTM being 1 - dimensional isboth integrable and involutive. Thus, M = L × M ′, where L is anull space of RadTM , which completes the proof.

Since both the RadTM and SM are integrable, there exists anatlas of local charts U ; u0 , u1, u2, . . . , un−1 such that ∂

∂u0 ∈

Γ(RadTM|U). Thus, the matrix of the degenerate tensor g on Mwith respect to the frames field ∂

∂uo, . . . , ∂

∂un−1 is

[g] =

[0 00 g

ij(uo, . . . , un−1)

], (25)

where gij = g(

∂∂ui, ∂∂uj

), det[gij] 6= 0 and ga1 = g1a = 0 for all

i, j ∈ 1, . . . , n − 1 and a ∈ 2, . . . , n + 1 . According to thegeneral transformations on a foliated manifold, we have

uo = uo(uo, u1, . . . , un+1), ui = ui(u1, . . . , un+1) .

Using above transformations, one can obtain an invariant field offrames on M , adapted to the decomposition (1).

Remark 1. It follows from Theorem 3 that S(TM) is integrablefor any choice of the coefficient D0 in (23). Therefore, for simplic-ity, we transform (23) such that D0 = 1. Moreover, Theorem 3along with Theorem 1 implies that the second fundamental form ofS(TM) and the shape operator AN are symmetric on Γ(S(TM)).

Let (M ′, g′) be a spacelike hypersurface of M , with induced Rie-mannian metric g′. Thus, with respect to a basis ξ, W1, . . . ,Wn−1 for TM , g and g′ are related by

g(X, Y ) =n−1∑a=1

g(X, Wa) g(Y, Wa) = g′(X, Y ), X, Y ∈ Γ(S(TM)).

Theorem 4. Let (M, g, S(TM)) be an n - dimensional half-lightlikesubmanifold of a Minkowski spacetime Rn+2

1 , with a canonical in-tegrable screen distribution S(TM) and (M ′, g′) a Riemannian hy-persurface of M such that M ′ is a leaf of S(TM). Then, M is(a) Ricci flat(b) totally geodesic(c) totally umbilical(d) minimalif and only if M ′ is so immersed as a submanifold of Rn+2

1 .

Proof. We know from the Remark 1 that the degenerate tensor gcan be identified with the Riemannian tensor g′ of M ′. Therefore,M and M ′ will have the same first fundamental forms. For a relationbetween their second fundamental forms, using (7), (20) and (24),for all X ∈ Γ(TM), we get,

ρ(X) = g(∇X N, ξ1 ) =1

2g(∇X ξ1, ξ1 ) = 0 ,

τ(X) = g(∇X N, ξ2 ) =1

2g(∇X ξ1, ξ2 )

= − 1

2g(Aξ1 X, ξ2) = 0

since Aξ1 X belongs to S(TM). Hence, (7) and (16) reduce to

∇X N = −AN X , ∇X ξ1 = −Aξ1 X. (26)

Now using D1(X, ξ1) = 0 from (9), (13) and (24), we obtain

∇X N =1

2∇X ξ1 =

1

2∇X ξ1 = − 1

2(Aξ1 X +D2 (X, ξ1)ξ2 ) (27)

for all X ∈ Γ(TM). The equations (13), (26) and (27) imply

AN X =1

2(Aξ1 X + θ(X) ξ2 ). (28)

On the other hand, the respective second fundamental forms andtheir shape operators are related by

g(D1(X, Y, N) = g(AN X, Y ), g(E(X, Y, ξ1) = g(Aξ1 X, Y ).

Using above in (28), for X ∈ Γ(TM), PY ∈ Γ(S(TM)), we get

g(AN X, PY ) =1

2g(Aξ1 X, PY ) +

1

2θ(X) g(ξ2, PY ).

As a consequence of the above relation and g(ξ2, PY ) = 0, we get

E(X, PY ) =1

2D1(X, PY ). (29)

Denote by h′ the second fundamental form of M ′. Then,

∇X Y = ∇′X Y + h′(X, Y ) , for all X, Y ∈ Γ(TM ′) ,

where ∇′ is a linear connection on M ′. By using Gauss equations(4), (6) and (15) and (29), for X, Y ∈ Γ(S(TM)), we obtain

∇X Y = ∇′X Y +D1(X, Y ) (1

2ξ1 +N) +D2(X, Y ) ξ2

= ∇′X Y +1

2D1(X, Y ) ξ1 + h(X, Y ).

Hence, we have the following relation between the second funda-mental forms of M and M ′:

h′(X, Y ) =1

2D1(X, Y ) ξ1 + h(X, Y ), (30)

where X, Y ∈ Γ(S(TM)). The result (30) and the Remark 1 implythat the geometries of M and M ′, directly related to the first andsecond fundamental forms, are same. Precisely, this means that(a)− (d) holds, which completes the proof.

In particular, as explained in Example 1, null 2-surfaces of R41

are examples for which Theorem 4 will hold. Also, see Example 2of [6, page 159] of a totally umbilical half-lightlike surface.

Remark 2. In case D1 and D2 vanish on M , then it follows fromTheorem 2 and (30) that h′ = h = 0, that is, M and M ′ are totallygeodesic in Rn+2

1 . Also, for this case, ∇ and ∇′ are metric (Levi -Civita) connections on M and M ′ respectively.

Remark 3. If the Minkowski space is replaced by an arbitraryLorentz manifold, then there is no guarantee for the existence of in-tegrable screen distribution of its half-lightlike submanifold. Thus,the results do not hold for an arbitrary ambient Lorentz manifold.However, we now show that there do exist physically significantLorentzian manifolds for which Theorem 4 will hold.

Well-known theorem of Geroch (see details in [7]) guaranteesthat there exists a globally hyperbolic spacetime, as a topologicalproduct R × S, where S is a Cauchy hypersurface. Such a space-time is of the form (M = R × B, g = −dt2 ⊕ Bg) with (B, Bg)a Riemannian manifold. Minkowski, De-Sitter and the Einsteinstatic universe are examples (see [1] for details). Furthermore,according to Beem and Ehrlich [1], there exists a class of glob-ally hyperbolic spacetimes of the form (M = R ×f B, g), wheref is a warped function. Among several examples, we mention 4-dimensional Robertson-Walker spacetimes for which the metric gcan be scaled down to the form ds2 = −dt2 + f(t)dΩ2. All suchproduct spacetimes possess at least one timelike covariant constantvector field, say V . In particular, choosing V = − ∂

∂t, using (22)

and (23) and then proceeding exactly as before, one can prove thatTheorem 4 will hold for any of above mentioned warped productglobally hyperbolic spacetimes. In particular, following Example 1,one can construct null 2-surfaces of Robertson-Walker spacetimes.For physical significance of globally hyperbolic spacetimes and a va-riety of other examples, which may satisfy Theorem 4, we refer [1].

Remark 4. Theorem 4 is important as it provides a way to reduce,for a class of half-lightlike manifolds, problems of null geometry toproblems of Riemannian geometry. In particular, one can study anyof the four types in Theorem 4 of half lightlike submanifolds by usingthe corresponding known geometry of its Riemannian hypersurface.Related to this problem we mention [3, 4] in which the presentauthor has used this idea and studied some curvature properties ofa class of null manifolds.

Finally, we propose the following open problem:Let (M, g) be an n-dimensional half-lightlike submanifold of an

(n + 2)-dimensional semi - Riemannian manifold (M, g). SupposeM has an integrable screen distribution S(TM). Find a class of Mfor which Theorem 4 holds.

References

[1] Beem, J. K., Ehrlich, P.E. and Easley, K.L., Global LorentzianGeometry, Marcel Dekker, New York, Second Edition, 1996.

[2] Berger, M., Riemannian Geometry During the Second Half ofthe Twentieth Century, Univ. Lecture Ser., Vol. 17, Amer.Math. Soc., 2000.

[3] Duggal, K. L., Constant scalar curvature and warped prod-uct globally null manifolds, Journal of geometry and physics,43(2002), 327-340.

[4] Duggal, K. L., Warped product of lightlike manifolds, Nonlin-ear Anal., 47(2001), 3061-3072.

[5] Duggal, K. L. and Bejancu, A., Lightlike submanifolds of codi-mension two, Math. J. Toyama Univ., 15(1992), 59-82.

[6] Duggal, K. L. and Jin, D. H., Half lightlike submanifolds ofcodimension 2, Math. J. Toyama Univ., 22(1999), 121-161.

[7] Hawking, S. W. and Ellis, G. F. R., The large scale structureof spacetime, Cambridge University Press, Cambridge, 1973.

K. L. DuggalDepartment of Mathematics and StatisticsUniversity of WindsorWindsor, Ontario, Canada N9B3P4.E-mail: yq8@uwindsor.ca